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Metamath Proof Explorer

Theorem tz6.12c

Description: Corollary of Theorem 6.12(1) of TakeutiZaring p. 27. (Contributed by NM, 30-Apr-2004) (Proof shortened by SN, 23-Dec-2024)

Ref Expression
Assertion tz6.12c ⊢ ∃! y A F y → F ⁡ A = y ↔ A F y

Proof

Step Hyp Ref Expression
1 df-fv ⊢ F ⁡ A = ι y | A F y
2 1 eqeq1i ⊢ F ⁡ A = y ↔ ι y | A F y = y
3 iota1 ⊢ ∃! y A F y → A F y ↔ ι y | A F y = y
4 2 3 bitr4id ⊢ ∃! y A F y → F ⁡ A = y ↔ A F y